1
Pendulum Averaging
Averaging the Rapidly Forced Pendulum
In these notes we find the averaged equation for
t
θ′′ + (a + f ( )F (θ) = 0,
ǫ
(1)
the equation for a rapidly forced pendulum. We assume that f is periodic, f (t) = f (t + 1),
and it has zero average,
Z 1
f=
f (t)dt = 0.
(2)
0
First we write the governing equation as a system,
t
′
′
θ = v,
v = − a + f ( ) F (θ).
ǫ
Now we make a change of time scale
t
ǫ
(3)
→ τ so that the governing system becomes
θ′ = ǫv,
v ′ = −ǫ (a + f (τ )) F (θ).
(4)
We seek a change of variables
θ = φ + ǫχ1 (φ, w, τ ) + ǫ2 χ2 (φ, w, τ ) + · · · ,
v = w + ǫW1 (φ, w, τ ) + ǫ2 W2 (φ, w, τ ) + · · · ,
(5)
(6)
in order to convert to the new system of equations
φ′ = ǫH0 (φ, w) + ǫ2 H1 (φ, w) + · · · ,
w ′ = ǫG0 (φ, w) + ǫ2 G1 (φ, w) + · · · .
(7)
We proceed by direct calculation, but keep it simple at first: To leading order in ǫ,
θ′ = φ′ + χ1τ (φ, w, τ ) = w,
(8)
χ1τ (φ, w, τ ) = w − H0 (φ, w),
(9)
so that
It follows that
χ1 = 0,
H0 (φ, w) = w.
(10)
Similarly,
v ′ = w ′ + W1τ (φ, w, τ ) = − (a + f (τ )) F (φ)
(11)
W1τ (φ, w, τ ) = − (a + f (τ )) F (φ) − G0 (φ, w).
(12)
so that
It follows that
W1 = −hF (φ),
G0 = −aF (φ),
h′ = f (τ ),
So far, so good. However, we have learned nothing useful.
1
h = 0.
(13)
Now to next order (order ǫ2 ):
θ = φ + ǫ2 χ2 (φ, w, τ ) + · · · ,
v = w − ǫh(τ )F (φ) + ǫ2 W2 (φ, w, τ ) + · · · ,
(14)
(15)
θ′ = φ′ + ǫχ2τ (φ, w, τ ) = w − ǫh(τ )F (φ),
v ′ = w ′ − h′ (τ )F (φ) − ǫh(τ )F ′ (φ)φ′ + ǫW2τ (φ, w) = − (a + f (τ )) F (φ),
(16)
(17)
χ2τ (φ, w, τ ) = −h(τ )F (φ) − H1 ,
W2τ (φ, w, τ ) = h(τ )F ′ (φ)w − G1 .
(18)
(19)
so that
and
It follows that
χ2 (φ, w, τ ) = −g(τ )F (φ),
W2 (φ, w, τ ) = g(τ )F ′(φ)w,
g ′ = h,
H1 = 0,
G1 = 0.
g = 0,
(20)
(21)
Okay, so this is not enough. We need to go to yet another order:
θ = φ − ǫ2 g(τ )F (φ) + ǫ3 χ3 (φ, w, τ ),
v = w − ǫh(τ )F (φ) + ǫ2 g(τ )F ′ (φ)w + ǫ3 W3 (φ, w, τ ) + · · · .
(22)
(23)
Differentiate wrt t to find
χ3τ (φ, w, τ ) = 2g(τ )F ′(φ)w − H2 ,
W3τ (φ, w, τ ) = g(τ )f (τ )F ′ (φ)F (φ) + g(τ )F ′′ (φ)w 2 − G2 .
(24)
(25)
It follows that
G2 = gf F ′ (φ)F (φ),
H2 = 0,
(26)
and the differential equation that results is
φ′ = w,
w ′ = −aF (φ) + ǫ2 gf F ′ (φ)F (φ),
(27)
Notice that
fg =
Z
1
f (t)g(t)dt =
0
Z
1
′
h (t)g(t)dt = −
0
Z
1
h2 (t)dt = −h2 .
(28)
0
So
φ′′ = −aF (φ) − ǫ2 h2 F ′ (φ)F (φ).
1.1
(29)
Specific example
Take F (φ) = sin(φ), and f (t) = α sin t. Then the averaged equation is
1
(30)
φ′′ = − a − ǫ2 α cos(φ) sin(φ).
2
Now the stability of the rest point at π is determined by the sign of −a − 21 ǫ2 α cos(π), and
the rest point is stable if
1
(31)
a < ǫ2 α
2
2